Polytope of Type {3,2,5,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,5,4}*1920
if this polytope has a name.
Group : SmallGroup(1920,240399)
Rank : 5
Schlafli Type : {3,2,5,4}
Number of vertices, edges, etc : 3, 3, 40, 80, 32
Order of s0s1s2s3s4 : 30
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,2,5,4}*960
16-fold quotients : {3,2,5,2}*120
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 4,164)( 5,174)( 6,177)( 7,171)( 8,168)( 9,178)( 10,173)( 11,167)
( 12,176)( 13,170)( 14,165)( 15,175)( 16,172)( 17,166)( 18,169)( 19,179)
( 20,228)( 21,238)( 22,241)( 23,235)( 24,232)( 25,242)( 26,237)( 27,231)
( 28,240)( 29,234)( 30,229)( 31,239)( 32,236)( 33,230)( 34,233)( 35,243)
( 36,212)( 37,222)( 38,225)( 39,219)( 40,216)( 41,226)( 42,221)( 43,215)
( 44,224)( 45,218)( 46,213)( 47,223)( 48,220)( 49,214)( 50,217)( 51,227)
( 52,196)( 53,206)( 54,209)( 55,203)( 56,200)( 57,210)( 58,205)( 59,199)
( 60,208)( 61,202)( 62,197)( 63,207)( 64,204)( 65,198)( 66,201)( 67,211)
( 68,180)( 69,190)( 70,193)( 71,187)( 72,184)( 73,194)( 74,189)( 75,183)
( 76,192)( 77,186)( 78,181)( 79,191)( 80,188)( 81,182)( 82,185)( 83,195)
( 84,244)( 85,254)( 86,257)( 87,251)( 88,248)( 89,258)( 90,253)( 91,247)
( 92,256)( 93,250)( 94,245)( 95,255)( 96,252)( 97,246)( 98,249)( 99,259)
(100,308)(101,318)(102,321)(103,315)(104,312)(105,322)(106,317)(107,311)
(108,320)(109,314)(110,309)(111,319)(112,316)(113,310)(114,313)(115,323)
(116,292)(117,302)(118,305)(119,299)(120,296)(121,306)(122,301)(123,295)
(124,304)(125,298)(126,293)(127,303)(128,300)(129,294)(130,297)(131,307)
(132,276)(133,286)(134,289)(135,283)(136,280)(137,290)(138,285)(139,279)
(140,288)(141,282)(142,277)(143,287)(144,284)(145,278)(146,281)(147,291)
(148,260)(149,270)(150,273)(151,267)(152,264)(153,274)(154,269)(155,263)
(156,272)(157,266)(158,261)(159,271)(160,268)(161,262)(162,265)(163,275);;
s3 := ( 4,180)( 5,195)( 6,182)( 7,193)( 8,186)( 9,189)( 10,184)( 11,191)
( 12,190)( 13,185)( 14,188)( 15,187)( 16,192)( 17,183)( 18,194)( 19,181)
( 20,164)( 21,179)( 22,166)( 23,177)( 24,170)( 25,173)( 26,168)( 27,175)
( 28,174)( 29,169)( 30,172)( 31,171)( 32,176)( 33,167)( 34,178)( 35,165)
( 36,228)( 37,243)( 38,230)( 39,241)( 40,234)( 41,237)( 42,232)( 43,239)
( 44,238)( 45,233)( 46,236)( 47,235)( 48,240)( 49,231)( 50,242)( 51,229)
( 52,212)( 53,227)( 54,214)( 55,225)( 56,218)( 57,221)( 58,216)( 59,223)
( 60,222)( 61,217)( 62,220)( 63,219)( 64,224)( 65,215)( 66,226)( 67,213)
( 68,196)( 69,211)( 70,198)( 71,209)( 72,202)( 73,205)( 74,200)( 75,207)
( 76,206)( 77,201)( 78,204)( 79,203)( 80,208)( 81,199)( 82,210)( 83,197)
( 84,260)( 85,275)( 86,262)( 87,273)( 88,266)( 89,269)( 90,264)( 91,271)
( 92,270)( 93,265)( 94,268)( 95,267)( 96,272)( 97,263)( 98,274)( 99,261)
(100,244)(101,259)(102,246)(103,257)(104,250)(105,253)(106,248)(107,255)
(108,254)(109,249)(110,252)(111,251)(112,256)(113,247)(114,258)(115,245)
(116,308)(117,323)(118,310)(119,321)(120,314)(121,317)(122,312)(123,319)
(124,318)(125,313)(126,316)(127,315)(128,320)(129,311)(130,322)(131,309)
(132,292)(133,307)(134,294)(135,305)(136,298)(137,301)(138,296)(139,303)
(140,302)(141,297)(142,300)(143,299)(144,304)(145,295)(146,306)(147,293)
(148,276)(149,291)(150,278)(151,289)(152,282)(153,285)(154,280)(155,287)
(156,286)(157,281)(158,284)(159,283)(160,288)(161,279)(162,290)(163,277);;
s4 := ( 4, 88)( 5, 89)( 6, 90)( 7, 91)( 8, 84)( 9, 85)( 10, 86)( 11, 87)
( 12, 96)( 13, 97)( 14, 98)( 15, 99)( 16, 92)( 17, 93)( 18, 94)( 19, 95)
( 20,104)( 21,105)( 22,106)( 23,107)( 24,100)( 25,101)( 26,102)( 27,103)
( 28,112)( 29,113)( 30,114)( 31,115)( 32,108)( 33,109)( 34,110)( 35,111)
( 36,120)( 37,121)( 38,122)( 39,123)( 40,116)( 41,117)( 42,118)( 43,119)
( 44,128)( 45,129)( 46,130)( 47,131)( 48,124)( 49,125)( 50,126)( 51,127)
( 52,136)( 53,137)( 54,138)( 55,139)( 56,132)( 57,133)( 58,134)( 59,135)
( 60,144)( 61,145)( 62,146)( 63,147)( 64,140)( 65,141)( 66,142)( 67,143)
( 68,152)( 69,153)( 70,154)( 71,155)( 72,148)( 73,149)( 74,150)( 75,151)
( 76,160)( 77,161)( 78,162)( 79,163)( 80,156)( 81,157)( 82,158)( 83,159)
(164,248)(165,249)(166,250)(167,251)(168,244)(169,245)(170,246)(171,247)
(172,256)(173,257)(174,258)(175,259)(176,252)(177,253)(178,254)(179,255)
(180,264)(181,265)(182,266)(183,267)(184,260)(185,261)(186,262)(187,263)
(188,272)(189,273)(190,274)(191,275)(192,268)(193,269)(194,270)(195,271)
(196,280)(197,281)(198,282)(199,283)(200,276)(201,277)(202,278)(203,279)
(204,288)(205,289)(206,290)(207,291)(208,284)(209,285)(210,286)(211,287)
(212,296)(213,297)(214,298)(215,299)(216,292)(217,293)(218,294)(219,295)
(220,304)(221,305)(222,306)(223,307)(224,300)(225,301)(226,302)(227,303)
(228,312)(229,313)(230,314)(231,315)(232,308)(233,309)(234,310)(235,311)
(236,320)(237,321)(238,322)(239,323)(240,316)(241,317)(242,318)(243,319);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s0*s1*s0*s1*s0*s1, s3*s4*s3*s4*s3*s4*s3*s4,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(323)!(2,3);
s1 := Sym(323)!(1,2);
s2 := Sym(323)!( 4,164)( 5,174)( 6,177)( 7,171)( 8,168)( 9,178)( 10,173)
( 11,167)( 12,176)( 13,170)( 14,165)( 15,175)( 16,172)( 17,166)( 18,169)
( 19,179)( 20,228)( 21,238)( 22,241)( 23,235)( 24,232)( 25,242)( 26,237)
( 27,231)( 28,240)( 29,234)( 30,229)( 31,239)( 32,236)( 33,230)( 34,233)
( 35,243)( 36,212)( 37,222)( 38,225)( 39,219)( 40,216)( 41,226)( 42,221)
( 43,215)( 44,224)( 45,218)( 46,213)( 47,223)( 48,220)( 49,214)( 50,217)
( 51,227)( 52,196)( 53,206)( 54,209)( 55,203)( 56,200)( 57,210)( 58,205)
( 59,199)( 60,208)( 61,202)( 62,197)( 63,207)( 64,204)( 65,198)( 66,201)
( 67,211)( 68,180)( 69,190)( 70,193)( 71,187)( 72,184)( 73,194)( 74,189)
( 75,183)( 76,192)( 77,186)( 78,181)( 79,191)( 80,188)( 81,182)( 82,185)
( 83,195)( 84,244)( 85,254)( 86,257)( 87,251)( 88,248)( 89,258)( 90,253)
( 91,247)( 92,256)( 93,250)( 94,245)( 95,255)( 96,252)( 97,246)( 98,249)
( 99,259)(100,308)(101,318)(102,321)(103,315)(104,312)(105,322)(106,317)
(107,311)(108,320)(109,314)(110,309)(111,319)(112,316)(113,310)(114,313)
(115,323)(116,292)(117,302)(118,305)(119,299)(120,296)(121,306)(122,301)
(123,295)(124,304)(125,298)(126,293)(127,303)(128,300)(129,294)(130,297)
(131,307)(132,276)(133,286)(134,289)(135,283)(136,280)(137,290)(138,285)
(139,279)(140,288)(141,282)(142,277)(143,287)(144,284)(145,278)(146,281)
(147,291)(148,260)(149,270)(150,273)(151,267)(152,264)(153,274)(154,269)
(155,263)(156,272)(157,266)(158,261)(159,271)(160,268)(161,262)(162,265)
(163,275);
s3 := Sym(323)!( 4,180)( 5,195)( 6,182)( 7,193)( 8,186)( 9,189)( 10,184)
( 11,191)( 12,190)( 13,185)( 14,188)( 15,187)( 16,192)( 17,183)( 18,194)
( 19,181)( 20,164)( 21,179)( 22,166)( 23,177)( 24,170)( 25,173)( 26,168)
( 27,175)( 28,174)( 29,169)( 30,172)( 31,171)( 32,176)( 33,167)( 34,178)
( 35,165)( 36,228)( 37,243)( 38,230)( 39,241)( 40,234)( 41,237)( 42,232)
( 43,239)( 44,238)( 45,233)( 46,236)( 47,235)( 48,240)( 49,231)( 50,242)
( 51,229)( 52,212)( 53,227)( 54,214)( 55,225)( 56,218)( 57,221)( 58,216)
( 59,223)( 60,222)( 61,217)( 62,220)( 63,219)( 64,224)( 65,215)( 66,226)
( 67,213)( 68,196)( 69,211)( 70,198)( 71,209)( 72,202)( 73,205)( 74,200)
( 75,207)( 76,206)( 77,201)( 78,204)( 79,203)( 80,208)( 81,199)( 82,210)
( 83,197)( 84,260)( 85,275)( 86,262)( 87,273)( 88,266)( 89,269)( 90,264)
( 91,271)( 92,270)( 93,265)( 94,268)( 95,267)( 96,272)( 97,263)( 98,274)
( 99,261)(100,244)(101,259)(102,246)(103,257)(104,250)(105,253)(106,248)
(107,255)(108,254)(109,249)(110,252)(111,251)(112,256)(113,247)(114,258)
(115,245)(116,308)(117,323)(118,310)(119,321)(120,314)(121,317)(122,312)
(123,319)(124,318)(125,313)(126,316)(127,315)(128,320)(129,311)(130,322)
(131,309)(132,292)(133,307)(134,294)(135,305)(136,298)(137,301)(138,296)
(139,303)(140,302)(141,297)(142,300)(143,299)(144,304)(145,295)(146,306)
(147,293)(148,276)(149,291)(150,278)(151,289)(152,282)(153,285)(154,280)
(155,287)(156,286)(157,281)(158,284)(159,283)(160,288)(161,279)(162,290)
(163,277);
s4 := Sym(323)!( 4, 88)( 5, 89)( 6, 90)( 7, 91)( 8, 84)( 9, 85)( 10, 86)
( 11, 87)( 12, 96)( 13, 97)( 14, 98)( 15, 99)( 16, 92)( 17, 93)( 18, 94)
( 19, 95)( 20,104)( 21,105)( 22,106)( 23,107)( 24,100)( 25,101)( 26,102)
( 27,103)( 28,112)( 29,113)( 30,114)( 31,115)( 32,108)( 33,109)( 34,110)
( 35,111)( 36,120)( 37,121)( 38,122)( 39,123)( 40,116)( 41,117)( 42,118)
( 43,119)( 44,128)( 45,129)( 46,130)( 47,131)( 48,124)( 49,125)( 50,126)
( 51,127)( 52,136)( 53,137)( 54,138)( 55,139)( 56,132)( 57,133)( 58,134)
( 59,135)( 60,144)( 61,145)( 62,146)( 63,147)( 64,140)( 65,141)( 66,142)
( 67,143)( 68,152)( 69,153)( 70,154)( 71,155)( 72,148)( 73,149)( 74,150)
( 75,151)( 76,160)( 77,161)( 78,162)( 79,163)( 80,156)( 81,157)( 82,158)
( 83,159)(164,248)(165,249)(166,250)(167,251)(168,244)(169,245)(170,246)
(171,247)(172,256)(173,257)(174,258)(175,259)(176,252)(177,253)(178,254)
(179,255)(180,264)(181,265)(182,266)(183,267)(184,260)(185,261)(186,262)
(187,263)(188,272)(189,273)(190,274)(191,275)(192,268)(193,269)(194,270)
(195,271)(196,280)(197,281)(198,282)(199,283)(200,276)(201,277)(202,278)
(203,279)(204,288)(205,289)(206,290)(207,291)(208,284)(209,285)(210,286)
(211,287)(212,296)(213,297)(214,298)(215,299)(216,292)(217,293)(218,294)
(219,295)(220,304)(221,305)(222,306)(223,307)(224,300)(225,301)(226,302)
(227,303)(228,312)(229,313)(230,314)(231,315)(232,308)(233,309)(234,310)
(235,311)(236,320)(237,321)(238,322)(239,323)(240,316)(241,317)(242,318)
(243,319);
poly := sub<Sym(323)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3 >;
to this polytope